I have been working with phonon dispersion relations for a while now on the topic of metamaterials (phononic band gaps). However, I still do not feel that I have fully grasped how to interpret these disperion relation diagrams.
The frequency $\omega$ is plotted against the wave vector $k$, but how do I actually read it? Do I search for a frequency and look which modes are "(co)existing" at that frequency? Or do I pick a wave vector (a direction) and look which frequencies are allowed for these values of $k$? I can probably read it both ways, but where is cause and effect exactly?
Here's what I know: Let's assume a 2D case with a simple Brillouin Zone $\Gamma$-X-Y-$\Gamma$. The sections of the dispersion relation correspond to values of $k$, where $\Gamma$ denotes the point where $k$ is very small and the wavelength $\lambda$ is very large. Traveling along the x-axis is basically like traversing the edges of the Brillouin Zone, covering all possible directions of the wave vector.
Suppose, a dispersion branch for $\Gamma$-X has two possible frequencies. What is the "real world meaning" of that? Do both these modes exist at a certain excitation frequency?
Now assume there are two different branches that occur at the same frequency inside $\Gamma$-X. Does that make it any different than case 1 where the same branch has one frequency twice?